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Vectors in Two Dimensions

Let's start learning about vectors in two dimensions. A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In two dimensions (2-d), we specify the direction of a vector relative to some reference frame (i.e., coordinate system) ...

Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai’i to Moloka’i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey)

Vectors in Two Dimensions

A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors.

In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

The figure below shows such a graphical representation of a vector, using as an example the total displacement for the person walking in a city considered in the previous lessons. We shall use the notation that a boldface symbol, such as \(\mathbf{D}\), stands for a vector. Its magnitude is represented by the symbol in italics, \(D\), and its direction by \(\theta \).

Vectors in this Tutorial

In this tutorial, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector \(\mathbf{F}\), which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as \(F\), and the direction of the variable will be given by an angle \(\theta \).

A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.

A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle \(\text{29}\text{.1º}\) north of east.

On a graph a vector is shown. It is inclined at an angle theta equal to twenty nine point one degrees above the positive x axis. A protractor is shown to the right of the x axis to measure the angle. A ruler is also shown parallel to the vector to measure its length. The ruler shows that the length of the vector is ten point three units.

To describe the resultant vector for the person walking in a city considered in the figure above graphically, draw an arrow to represent the total displacement vector \(\mathbf{D}\). Using a protractor, draw a line at an angle \(\theta \) relative to the east-west axis. The length \(D\) of the arrow is proportional to the vector’s magnitude and is measured along the line with a ruler. In this example, the magnitude \(D\) of the vector is 10.3 units, and the direction \(\theta \) is \(29.1º\) north of east.

This lesson is part of:

Two-Dimensional Kinematics

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